一种最优幂律升级算法
An Upgrading Algorithm with Optimal Power Law
摘要：
考虑通道$W$以及给定的输入分布$P_X$。在某些设置中，诸如在极化码的构造中，$W$的输出字母表“太大”，因此我们用具有较小输出字母表的信道$Q$来替换$W$。如果通过处理$Q$的输出从$W$获得，我们说$Q$相对于$W$升级。在这种情况下，$W$的输入和输出之间的互信息$I(P_X,W)$由$Q$的输入和输出之间的互信息$I(P_X,Q)$限定上限。在本文中，我们提出了一种算法，从$W$产生一个升级的通道$Q$，作为$P_X$的函数和所需的输出字母大小的$Q$，表示为$L$。我们表明，在互信息的差异是“小”。也就是说，它是$O(L^{-2/(|\mathcal{X}|-1)})$，其中$|\mathcal{X}|$是输入字母表的大小。$L$的幂律是最优的。我们补充我们的分析与数值实验表明，所开发的算法改进后，现有的国家的最先进的算法也在非渐近设置。
英文摘要：
Consider a channel $W$ along with a given input distribution $P_X$. In certain settings, such as in the construction of polar codes, the output alphabet of $W$ is `too large', and hence we replace $W$ by a channel $Q$ having a smaller output alphabet. We say that $Q$ is upgraded with respect to $W$ if $W$ is obtained from $Q$ by processing its output. In this case, the mutual information $I(P_X,W)$ between the input and output of $W$ is upper-bounded by the mutual information $I(P_X,Q)$ between the input and output of $Q$. In this paper, we present an algorithm that produces an upgraded channel $Q$ from $W$, as a function of $P_X$ and the required output alphabet size of $Q$, denoted $L$. We show that the difference in mutual informations is `small'. Namely, it is $O(L^{-2/(|\mathcal{X}|-1)})$, where $|\mathcal{X}|$ is the size of the input alphabet. This power law of $L$ is optimal. We complement our analysis with numerical experiments which show that the developed algorithm improves upon the existing state-of-the-art algorithms also in non-asymptotic setups.
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